Godel is hailed as our second greatest logician because of his incompleteness theorems. We'll take a first peek at those theorems tomorrow—but for today, we tip our hat to Rebecca Goldstein.

Goldstein wrote the book on Godel; she did so in 2005. Along the way, she outlined Aristotle's work as a logician, in a footnote on page 55:

GOLDSTEIN (page 55): Aristotle is commonly acknowledged as the father of logic. His work in logic is laid out in the Prior Analytics, which is part of the posthumous consortium known as the Organon. The philosopher had the seminal insight that in a deductive logical argument, some words are logically relevant while others are not. The irrelevant words can be dispensed with by making them variables...

Goldstein goes on to offer a "stock syllogism" as an example of what she's talking about. She then says, "The move toward denoting logically irrelevant words with variables was a move toward generality and thus toward the science of logic."

At this point, we start in surprise. Has someone actually made a science of logic? If you read American newspapers, or if your watch our "cable news" programs, you might be greatly surprised by any such allegation.

As everyone must understand by now, "the science of logic" plays almost no role in our modern American discourse. Within our failing culture's pseudo-discussions, basic information and basic facts are almost wholly verboten. But anything resembling logic is typically frowned upon too.

This destructive state of affairs can't be blamed on Aristotle. Because Godel lived in modern times, we'll be a bit less forgiving when it comes to him.

Sad! Modern logicians have created a system in which their highly abstruse ruminations have nothing to do with the bungled logic of our public discussions. Godel lived until 1978. At some point, he probably should have stepped in.

At any rate, our modern logicians walk to work in the clouds, where they engage in high-brow endeavors which even they may not be equipped to explain. This brings us to a fascinating part of Godel's horrifically tragic story, as told by Goldstein in 2005 and then by Jim Holt in the derivative title essay of his well-received new book.

We refer to a possibly crazy idea to which Godel was apparently devoted throughout the course of his life. We refer to Godel's alleged belief in "Platonism," an alleged doctrine with which Goldstein says Godel "fell in love" when he was in his late teens.

In our previous report, we learned an unfortunate fact. Godel, who died of self-starvation, exhibited signs of severe mental illness throughout the course of his highly unusual life.

"At the age of five, he seems to have suffered a mild anxiety neurosis," Holt writes in his new book, drawing on Goldsetin's work. "At eight, he had a terrifying bout of rheumatic fever, which left him with the lifelong conviction that his heart had been fatally damaged."

By normal reckoning, Godel's apparent paranoia eventually led to his death. But then, as noted yesterday, he had always struck the Princeton community as being extremely odd.

In her book, Goldstein cites an array of peculiar ideas Godel is said to have expressed. Our continuing question is this:

Should it seem strange to think that the western world's second greatest logician was apparently in the grip of serious mental illness? Asking a slightly different question, should it seem strange that "Aristotle's successor" was in the grip, throughout his life, of some extremely peculiar ideas?

In theory, the mental illness, however tragic, shouldn't matter at all. If a person achieves some great intellectual breakthrough—in the physical sciences, let's say—that achievement isn't negated by the tragedy of a severe mental illness.

That said, Godel's achievements didn't come in the realm of physical science. Indeed, Goldstein quotes him telling one Princeton-based scholar, "I don't believe in natural science." She quotes him telling Thomas Nagel that he doesn't believe in evolution, offering Stalin's similar disbelief as supporting evidence.

She quotes him saying this to Noam Chomsky: "I am trying to prove that the laws of nature are a priori." Should it seem strange to hear that history's second greatest logician was bruiting such notions around?

In fairness, these peculiar-sounding statements were made during Godel's adult life. He formulated his "incompleteness theorems" when he was just 23.

If those theorems make sense and are actually important in some way, later delusions and mental illness can't vitiate the achievement. That said, we might want to consider the lifelong belief in "Platonism" which, according to Goldstein's book, played the key role in Godel's thinking from his teenage years on.

What the heck is Platonism? According to Goldstein, Godel "fell in love" with the doctrine while he was still a teen.

"First exposure to Plato can be an extremely heady experience for those with a passion for abstraction," Goldstein writes. "It can amount to a sort of ecstasy."

Parenthetically, Goldstein says that she can remember her own first exposure to Plato. But what is the doctrine which may result? What is this "Platonism?"

In the opening essay of Holt's new book, he defines this doctrine, or he at least pretends or attempts to do so. As critics robotically praise the clarity of Holt's writing, he sketches the doctrine like this:

HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians...

Say what? According to Holt, Godel, who was still in his teens, was "seduced by" a certain notion. More specifically (or possibly less so), he was seduced by "the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind."

According to Holt, this "doctrine," which is called Platonism, "has always been popular among mathematicians." In fairness, this suggests that it isn't something the teen-aged Godel somehow dreamed up by himself.

At this point, our review of the Godel file reached a critical turn. We're told that the western world's second greatest logician was swept away by a seductive idea, an idea which lay at the heart of his subsequent work.

What did this greatest logician believe? According to Holt, he believed that numbers have a perfect, timeless existence independent of the human mind. So Kurt Godel believed!

Readers, how about it? Do you believe that the number 3 "has a perfect existence independent of the human mind?" Do you believe that the number 3 has a perfect existence at all?

Indeed, let's ask the more relevant question: Do you have even the slightest idea what Holt is talking about? Granting the fact that you can memorize Teacher's words and reproduce those words on the test, would you have even the slightest idea what you were talking about?

Readers may be inclined to suppose that Holt proceeds from there to explain this murky bit of word salad. In our view, such a supposition would be badly mistaken.

Does Holt clarify this description? We'll set that question aside for another day. Instead, we'll turn to Goldstein, to the start of her attempt to explicate "Platonism."

Godel fell in love with the doctrine, she says. She lays out the doctrine as follows:

GOLDSTEIN (page 44): Godel's commitment to the objective existence of mathematical reality is the view known as conceptual, or mathematical realism. It is also known as mathematical Platonism, in honor of the ancient Greek philosopher...

Platonism is the view that the truths of mathematics are independent of any human activities, such as the construction of formal systems—with their axioms, definitions, rules of inference and proofs. The truths of mathematics are determined, according to Platonism, by the reality of mathematics, by the nature of the real, though abstract entities (numbers, sets, etc.) that make up that reality. The structure of, say, the natural numbers (which are the regular old counting numbers: 1, 2, 3, etc.) exists independent of us, according to the mathematical realist...and the properties of the numbers 4 and 25—that, for example, one is even, the other is odd and both are perfect squares—are as objective as are, according to the physical realist, the physical properties of light and gravity.

Goldstein muddies her account with a fair amount of technical language. (How many general interest readers are likely to know what a "formal system" is?)

Beyond that, she seems to think, all through her book, that the terms "objective" and "subjective" explain themselves in contexts like this. Sadly, they very much don't.

Whatever! In this passage, Goldstein starts describing the doctrine with which Godel "fell in love" as a teen. According to Goldstein, the ardor Godel felt for this doctrine impelled him to devise the "incompleteness theorems" which made him famous among certain academics, while leaving him completely unknown to everyone else on the face of the earth.

Godel's ardor for this alleged doctrine forms the core of Goldstein's work. And how does she describe the doctrine? She describes it like this:

According to Goldstein, Godel fell in love with the view that "the truths of mathematics are determined...by the reality of mathematics."

So far, so completely no good! Explicating further, she says Godel adopted the view that "the truths of mathematics are determined by the nature of the entities that make up" the reality of mathematics.

As an overview, she starts by saying that Godel developed a "commitment to the objective existence of mathematical reality!"

Readers, tell the truth. Would you say that the essence of this key doctrine has begun to come clear? We're going to say that we're munching on salad—that it's word salad all the way down!

At this point, we'll raise two questions, after which we'll adjourn for the day. One of our questions will concerns Kurt Godel. The other question will concern writers like Goldstein and Holt, who are robotically praised, by scripted journalists, for the clarity of their work.

Regarding Godel, we'll ask this question: Is it possible that the western world's second greatest logician fell in love at an early age with a crackpot idea?

He believed many other crazy things in the course of his tragic life. Could this idea, which drove the work for which he's lauded, possibly have been the first of his crackpot ideas?

That's a question about our second greatest logician. Our second question concerns Holt and Goldstein—and it's built on a basic concept:

Here's the basic concept. Some claims or ideas are right or wrong. But some claims, and some ideas, don't rise to that level.

Such claims and ideas aren't right or wrong—they're simply incoherent!

This idea is stunningly basic. To what extent are general interest writers like Goldstein and Holt conversant with this idea?

25 comments:

Suppose you were sending a message to an alien race on some other planet. You could include arithmetic statements like "1 + 1 = 2" because these numbers and concepts or addition and equality have truth independent of human activity.

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I will say that whenever I hear a Karen Carpenter song, which is rare nowadays (though I did hear one just a couple of days ago on one of our local, eclectic stations), I’m literally floored. Everything else falls away when I hear that voice.

I'd recommend Bob study some logic, as he'd then realize that his trope here rests on an equivocation over the term "logic." Godel worked in formal logic, Bob's complaint about the absence of logic in modern discourse relates to informal logic. What informal logic is (or whether one should call it that, rather than something else) is a much-debated question nowadays (one thing not included in this conversation is Godel--not because Godel isn't actually a great logician, but because it's practically a different subject).

"Could this idea, which drove the work for which he's lauded, possibly have been the first of his crackpot ideas?"

What is Somerby driving at? He ridicules Holt and Goldstein for their attempts to explain Gödel's so-called views, but then he implies that Gödel's own views were (*possibly*) "crackpot." And yet, "later delusions and mental illness can't vitiate the achievement" "if those theorems make sense and are actually important in some way." Say what? Is Somerby going to attempt to debunk Gödel? Is Gödel's alleged "severe mental illness" somehow important to Somerby's assessment of Gödel's work, at the same time he asserts that a true achievement, such as that in the "physical sciences", cannot be vitiated by "mental illness?"

I'm with you, @2:18. I don't know were Somerby is headed. Is he trying to refute Godel's incompleteness or, at least, to disparage its importance? Well, maybe future posts will make Bob's position more clear.

Condemning Gödel's embrace of aspects of Platonism as possibly "crackpot" betrays a possible misrepresentation, misunderstanding, or oversimplification by Holt and Goldstein, and Somerby as well. Somerby seems to accept or promote the notion that Gödel was mentally ill to call Gödel's Platonism into question, but Somerby can't refrain from a flippant disregard for the questions Gödel was thinking about that led him to a Platonist view of mathematics.

Aside from studying logic, as one commenter above suggested, Somerby probably needs to study some mathematics if he wants to grapple intelligently with Gödel.

Not being an expert, I'm not entirely, totally, completely convinced that there isn't some foundational flaw in the work on mathematical logic of either Russell or Gödel. Are the questions that they addressed in some sense ill-posed? I have some nagging concerns, although I'd have a hard time articulating them with any clarity. That said, if we want to put mathematics on a rigorous foundation, we would at least have to understand how the questions addressed by these thinkers are ill-posed.

Might Plato also have been asking ill-posed questions? I know even less, but I'm always very leery of any question that asks whether something "exists"---that seems to be the most slippery word in all of philosophy, so, for myself, I'd be much more open to the idea that Plato barking in the wrong cave.

However, if Plato's questions were ill-posed, and if Gödel was much taken with Plato, does that imply that Gödel's work on incompleteness was also ill-posed? Not necessarily....I'd even venture to say NO. There is no clear line that I can see from Plato to incompleteness...I don't see how the latter is contingent on the former.

Similarly, Einstein was much taken with the ideas of Ernst Mach. I'd go so far as to say that Mach's ideas were wrong, but that in no way implies that General Relativity is wrong. An wrong idea can play a role in encouraging a thinker to do work that is correct and even great.

Hey Bob, when you retire why not take a few college classes -- on any serious subject! (No, psychology and journalism don't count.) How about some math beyond calculus; maybe some classes dealing with "algebra" (which you may be surprised to find out isn't something you already learned in high school.) The physical sciences -- even the "old" stuff from before the 20th century -- require some training in mathematics in order to be understood. No, it isn't fair. Sorry, not everything can be grasped by amateurs and the self-educated, people who always quit and move on to something else when the going gets hard. I know this must sound weird in an age when everyone feels entitled to know the Mind of God -- but please learn something about what you're criticizing!

Readers, tell the truth. Would you say that the essence of this key doctrine has begun to come clear?

To tell the truth, for this reader, yes. But perhaps because this reader has familiarized himself with the terminology and doesn’t expect every book to start with first principles. Also because the topic isn’t that difficult.

We’re all familiar with three-sided figures. We can draw them by putting pen to paper or arrange them out of sticks. At some point it’s natural to wonder what all these physical figures have in common, what’s the essence of a three-sided figure. We can think of the perfect triangle: three non-collinear, co-planar straight lines of no width (call them sides) that intersect pairwise in distinct points of no width and no length (call them vertices). All the dross three-sided figures of our experience are imperfect approximations: no matter how narrow our nib, our drawn lines always have some thickness, and no matter how true our lumber, our carpentered sides aren’t perfectly straight.

We can prove exact statements about the perfect three-sided figure, which we’ll call a triangle. The sum of the lengths of any two sides of any triangle is strictly larger than the length of the third side. The sum of the angle measures of the angles in a triangle is exactly 180 degrees. The area enclosed by any triangle is exactly one half the length of any side multiplied by the length of a perpendicular from the vertex opposite that side to that side.

These properties for three-sided objects are only approximate, in part because different subjects who measure these objects will record slightly different measurements and in part because the objects cannot be perfect. Not so for triangles: the essence of these objects does not vary, and they are not constrained by the imperfect tools of those subjects who contemplate them.

Most people are willing to concede the existence of a world outside ourselves. (Except the solipsists, and they don’t believe the rest of us are here anyway.) So most people would be willing to agree that if three trees in a Permian forest had fallen in such a way so as to create a three-sided figure, then that figure would have existed. Even though it would have been millions of years before any consciousness could have considered the scene.

But what about triangles? Did human minds invent them or discover them? If the former, then they have no existence beyond our minds. If the latter, then these are objects they don’t depend on our necessarily-subjective consciousness, and their properties (like the area formula mentioned above) also exist independently. In that case, the truth of the area formula is a truth of mathematics (or here, plane geometry) and is determined by the reality of triangles. That truth existed at the Big Bang and will exist after the heat death of this universe.

Now, you may reasonably decide that whether a reified abstraction exists is merely a matter of semantics, but that doesn’t mean you have to be a crackpot to contemplate the matter. Some ideas are right or wrong, and some of those will be incoherent to an ignoramus.

Gödel considered the abstraction not of geometric figures, but of systems of logic, i.e., the language and assumptions that we use to decide whether statements of the language are true or false). Gödel wanted to know whether we can use logical machinery to prove that the system is consistent (i.e, that it cannot lead to a contradiction) and whether we can use that machinery to prove any true statement of the system.

I can’t wait to see what the slowest boy in the class has to say about all this tomorrow.

Triangles, shmiangles deadrat. As they exist in plane geometry is a far cry from their existence in the physical world, which we perceive through our imperfect perceptions, and grasp upon to make into avatars – that is, perfect objects, which have certainly existed in the form of art long before their mathematical properties were ever realized. Hey, maybe minerals can form perfect triangles in the form of crystals, but they just aren’t the same as triangles in plane geometry. Rather, they would rely on the abstractions of molecular or atomic means of mathematical theory to explain how they came into being.

And the numbers that the great theorists manipulate to explain phenomena at that level have a certain elegance which can predict with an amazing degree of accuracy. But even today, Newton’s theory of gravity is still the go-to for NASA when calculating the trajectory of heavenly bodies and spacecraft, despite the refinements of general relativity to his theory.

“Gödel considered the abstraction not of geometric figures, but of systems of logic, i.e., the language and assumptions that we use to decide whether statements of the language are true or false.”

But wasn’t that in mathematical systems of logic? Because when it comes to language, all bets are off, far as I’m concerned. You might be able to prove the statement “Whatever I say is true” is nonsense, but only if you can disprove it. And I’m not sure mathematics provides that toolbox.

Triangles, shmiangles deadrat. As they exist in plane geometry is a far cry from their existence in the physical world

That sounds awfully familiar. Almost if I’d read it a short time ago. Or written it.

and grasp upon to make into avatars – that is, perfect objects

A Platonist would say you have it backwards. It’s not that we abstract our real-world experience to make avatars, but rather it’s that the avatars have always existed in the Platonic realm beyond our direct perception. The avatars project themselves into our world as imperfect shadows of their glorious, perfect selves.

Rather, they would rely on the abstractions of molecular or atomic means of mathematical theory to explain how they came into being.

I’m not sure of the antecedent of “they”, but Einstein put paid to the claim that molecules and atoms are mere abstractions, useful only to compute things in statistical mechanics. That was in 1905.

And the numbers that the great theorists manipulate to explain phenomena at that level have a certain elegance which can predict with an amazing degree of accuracy.

Sure, but do they exist a priori or did we make them up to make predictions?

But wasn’t that in mathematical systems of logic?

Absolutely.

You might be able to prove the statement “Whatever I say is true” is nonsense, but only if you can disprove it

Well, I can prove false the statement “Whatever I say is true” if David in Cal says it. Will that do?

I’m busted. I did smoke some doobies, but quite a long time ago. Luckily, I’m not running for the SCOTUS.

It seems I’ve been the victim of unclear perceptions in the realm of reader comprehension, which is a genetic aberration (laziness genes). As a result, I read too fast, without sussing out the entire point you were trying to make, and only glommed on to a reason to debate.

I was clumsily trying to point out that logic in mathematics does not translate very well to the emergent properties of the universe, of which we’re a part. And it’s only through agreement, through reason between us, as human beings, that the truth can be accepted as such.

Perhaps mathematics can play a role in predicting how masses of individually sentient beings will react to any given situation. With individuals, another matter. But it’s a sure bet that when such beings consume the same information, they will act in concert to the stimulus, though how that manifests itself is always uncertain.

Oxygen, meet hydrogen!

Sorry, it’s raining here, and I had a flashback to my doobie days, in which I encountered the postulation that even atoms may be, in a way, sentient. They always “know” how to react.

To get on with my long-winded point, I did not consider your observations on Platonism. I can only say that certain things, abstract or not, certainly existed before we ever emerged from the matter which defines us. In that sense, Plato was surely on point. My best spiritual estimate is that we are the universe trying to understand itself. And we seem to think we know how it will go. With our limited window, we only see beginnings and endings, so we’re a bit biased in that direction of thinking.

For we creatures, the evidence does not lie. Did Plato see death as an abstract truth? Is there an eventual heat death of the universe? Who knows!

“Some ideas are right or wrong, and some of those will be incoherent to an ignoramus.”

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